Can I use these both interchangeably?
You look just like I did at your age.
You look just like I was at your age.

Question

Answer

"You look just like me when I was your age." NOT "You look just like I was at your age."
"You look just like I did at your age" is correct and means the same thing.

Answer

Yes, you can. However in this case, "You look just like I did at your age"  would make more sense.

Another example might be, "You play sports just like I did at your age" or "You are tall just like I was at your age."